Language Of Arithmetic Research Articles

Hyperarithmetical complexity of infinitary action logic with multiplexing

Abstract In 2023, Kuznetsov and Speranski introduced infinitary action logic with multiplexing $!^{m}\nabla \textrm{ACT}_{\omega }$ and proved that the derivability problem for it lies between the $\omega $ level and the $\omega ^{\omega }$ level of the hyperarithmetical hierarchy. We prove that this problem is $\varDelta ^{0}_{\omega ^{\omega }}$-complete under Turing reductions. Namely, we show that it is recursively isomorphic to the satisfaction predicate for computable infinitary formulas of rank less than $\omega ^{\omega }$ in the language of arithmetic. As a consequence we prove that the closure ordinal for $!^{m}\nabla \textrm{ACT}_{\omega }$ equals $\omega ^{\omega }$. We also prove that the fragment of $!^{m}\nabla \textrm{ACT}_{\omega }$ where Kleene star is not allowed to be in the scope of the subexponential is $\varDelta ^{0}_{\omega ^{\omega }}$-complete. Finally, we present a family of logics, which are fragments of $!^{m}\nabla \textrm{ACT}_{\omega }$, such that the complexity of the $k$-th logic lies between $\varDelta ^{0}_{\omega ^{k}}$ and $\varDelta ^{0}_{\omega ^{k+1}}$.

Bilingual problem size effect: an ERP study of multiplication verification and production in two languages.

The problem size effect (PSE) is defined by better performance solving small problems (e.g., 2x4) than large problems (e.g., 8x9). For monolinguals, the PSE is larger when problems are presented in unfamiliar formats (e.g., written words), reflecting increased processing difficulty. Bilinguals are typically faster and more accurate at retrieving multiplication facts in the language of learning (LA+) than in their other language (LA-). We hypothesized that the less familiar arithmetic language (i.e., LA-) would elicit larger PSEs than LA+. Here, fluent Spanish-English bilingual adults verified spoken multiplication problems presented in LA+ and LA- while event-related potentials (ERPs) were recorded (Experiment 1A). To further promote language differences, we increased task difficulty by presenting problems at a faster pace (Experiment 1B) and requiring bilinguals to verbally produce solutions (Experiment 2). Language differences in performance were only observed for Experiment 2, where solutions were produced more slowly in LA- than LA+. In the ERPs, a PSE was driven by larger P300s for small than large solutions. A language effect was only observed under time pressure where LA- elicited a PSE at the 2nd operand. Additionally, the PSE was smaller for LA- at the solution. This suggests that categorizing multiplication facts is more effortful in LA-. In sum, very subtle language differences arise in fluent bilinguals when problems are more difficult, such as larger problems presented under time pressure in a weaker language. Critically, the effect of LA+ is at the level of response production and not access to the facts from memory.

Implicit commitment in a general setting

Abstract Gödel’s Incompleteness Theorems suggest that no single formal system can capture the entirety of one’s mathematical beliefs, while pointing at a hierarchy of systems of increasing logical strength that make progressively more explicit those implicit assumptions. This notion of implicit commitment motivates directly or indirectly several research programmes in logic and the foundations of mathematics; yet there hasn’t been a direct logical analysis of the notion of implicit commitment itself. In a recent paper, we carried out an initial assessment of this project by studying necessary conditions for implicit commitments; from seemingly weak assumptions on implicit commitments of an arithmetical system $S$, it can be derived that a uniform reflection principle for $S$—stating that all numerical instances of theorems of $S$ are true—must be contained in $S$’s implicit commitments. This study gave rise to unexplored research avenues and open questions. This paper addresses the main ones. We generalize this basic framework for implicit commitments along two dimensions: in terms of iterations of the basic implicit commitment operator, and via a study of implicit commitments of theories in arbitrary first-order languages, not only couched in an arithmetical language.

Modern perspectives in Proof Theory.

Modern perspectives in Proof Theory.

MODEL THEORY AND PROOF THEORY OF THE GLOBAL REFLECTION PRINCIPLE

AbstractThe current paper studies the formal properties of the Global Reflection Principle, to wit the assertion “All theorems of$\mathrm {Th}$are true,” where$\mathrm {Th}$is a theory in the language of arithmetic and the truth predicate satisfies the usual Tarskian inductive conditions for formulae in the language of arithmetic. We fix the gap in Kotlarski’s proof from [15], showing that the Global Reflection Principle for Peano Arithmetic is provable in the theory of compositional truth with bounded induction only ($\mathrm {CT}_0$). Furthermore, we extend the above result showing that$\Sigma _1$-uniform reflection over a theory of uniform Tarski biconditionals ($\mathrm {UTB}^-$) is provable in$\mathrm {CT}_0$, thus answering the question of Beklemishev and Pakhomov [2]. Finally, we introduce the notion of a prolongable satisfaction class and use it to study the structure of models of$\mathrm {CT}_0$. In particular, we provide a new model-theoretical characterization of theories of finite iterations of uniform reflection and present a new proof characterizing the arithmetical consequences of$\mathrm {CT}_0$.

ORDER TYPES OF MODELS OF FRAGMENTS OF PEANO ARITHMETIC

AbstractThe complete characterisation of order types of non-standard models of Peano arithmetic and its extensions is a famous open problem. In this paper, we consider subtheories of Peano arithmetic (both with and without induction), in particular, theories formulated in proper fragments of the full language of arithmetic. We study the order types of their non-standard models and separate all considered theories via their possible order types. We compare the theories with and without induction and observe that the theories without induction tend to have an algebraic character that allows model constructions by closing a model under the relevant algebraic operations.

ON THE UNCOUNTABILITY OF

AbstractCantor’s first set theory paper (1874) establishes the uncountability of ${\mathbb R}$ . We study this most basic mathematical fact formulated in the language of higher-order arithmetic. In particular, we investigate the logical and computational properties of ${\mathsf {NIN}}$ (resp. ${\mathsf {NBI}}$ ), i.e., the third-order statement there is no injection resp. bijection from $[0,1]$ to ${\mathbb N}$ . Working in Kohlenbach’s higher-order Reverse Mathematics, we show that ${\mathsf {NIN}}$ and ${\mathsf {NBI}}$ are hard to prove in terms of (conventional) comprehension axioms, while many basic theorems, like Arzelà’s convergence theorem for the Riemann integral (1885), are shown to imply ${\mathsf {NIN}}$ and/or ${\mathsf {NBI}}$ . Working in Kleene’s higher-order computability theory based on S1–S9, we show that the following fourth-order process based on ${\mathsf {NIN}}$ is similarly hard to compute: for a given $[0,1]\rightarrow {\mathbb N}$ -function, find reals in the unit interval that map to the same natural number.

Representations and the Foundations of Mathematics

The representation of mathematical objects in terms of (more) basic ones is part and parcel of (the foundations of) mathematics. In the usual foundations of mathematics, i.e. $\textsf{ZFC}$ set theory, all mathematical objects are represented by sets, while ordinary, i.e. non-set theoretic, mathematics is represented in the more parsimonious language of second-order arithmetic. This paper deals with the latter representation for the rather basic case of continuous functions on the reals and Baire space. We show that the logical strength of basic theorems named after Tietze, Heine, and Weierstrass, changes significantly upon the replacement of 'second-order representations' to 'third-order functions'. We discuss the implications and connections to the Reverse Mathematics program and its foundational claims regarding predicativist mathematics and Hilbert's program for the foundations of mathematics. Finally, we identify the problem caused by representations of continuous functions and formulate a criterion to avoid problematic codings within the bigger picture of representations.

Reflection algebras and conservation results for theories of iterated truth

We consider extensions of the language of Peano arithmetic by transfinitely iterated truth definitions satisfying uniform Tarskian biconditionals. Without further axioms, such theories are known to be conservative extensions of the original system of arithmetic. Much stronger systems, however, are obtained by adding either induction axioms or reflection axioms on top of them. Theories of this kind can interpret some well-known predicatively reducible fragments of second-order arithmetic such as iterated arithmetical comprehension.We obtain sharp results on the proof-theoretic strength of these systems using methods of provability logic. Reflection principles naturally define unary operators acting on the semilattice of axiomatizable extensions of our basic theory of iterated truth. The substructure generated by the top element of this algebra provides a canonical ordinal notation system for the class of theories under investigation.Using these notations we obtain conservativity relationships for iterated reflection principles of different logical complexity levels corresponding to the levels of the hyperarithmetical hierarchy, i.e., the analogs of Schmerl's formulas. These relationships, in turn, provide proof-theoretic analysis of our systems and of some related predicatively reducible theories. In particular, we uniformly calculate the ordinals characterizing the standard measures of their proof-theoretic strength, such as provable well-orderings, classes of provably recursive functions, and Π10-ordinals.

Илья Федорович Копиевский (Жизнь и судьба одного просветителя петровского времени)

In 1699, on the initiative of Peter the Great, the printing of Russian secular books began in Amsterdam, most of which were textbooks: on history, arithmetic, astronomy, navigation, and foreign languages. The compiler, translator, and publisher of these books was a native of the lands of the Grand Duchy of Lithuania, Ilya Fedorovich Kopievsky (or Kopievich). For many decades, historians have turned to the biography of this man, but it is still full of gaps and factual errors. The article summarizing the various information about Kopievsky available today (from archival documents to the latest works of historians) contains a detailed reconstruction of his life path. It also includes materials to return to the question of the contribution of this enlightener to the cultural reforms of Peter the Great.

Not all Kripke models of [formula omitted] are locally [formula omitted

Let K be an arbitrary Kripke model of Heyting Arithmetic, HA. For every node k in K, we can view the classical structure of k, Mk as a model of some classical theory of arithmetic. Let T be a classical theory in the language of arithmetic. We say K is locally T, iff for every k in K, Mk⊨T. One of the most important problems in the model theory of HA is the following question: Is every Kripke model ofHAlocallyPA? We answer this question negatively. We introduce two new Kripke model constructions to this end. The first construction actually characterizes the arithmetical structures that can be the root of a Kripke model K⊩HA+ECT0 (ECT0 stands for Extended Church Thesis). The characterization says that for every arithmetical structure M, there exists a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr=M iff M⊨ThΠ2(PA). One of the consequences of this characterization is that there is a rooted Kripke model K⊩HA+ECT0 with the root r such that Mr⊭IΔ1 and hence K is not even locally IΔ1. The second Kripke model construction is an implicit way of doing the first construction which works for any reasonable consistent intuitionistic arithmetical theory T with a recursively enumerable set of axioms that has the existence property. We get a sufficient condition from this construction that describes when for an arithmetical structure M, there exists a rooted Kripke model K⊩T with the root r such that Mr=M. As applications of this sufficient condition, we construct two new Kripke models. The first one is a Kripke model K⊩HA+¬θ+MP (θ is an instance of ECT0 and MP is Markov's principle) which is not locally IΔ1. The second one is a Kripke model K⊩HA such that K forces exactly the sentences that are provable from HA, but it is not locally IΔ1. Also, we will prove that every countable Kripke model of intuitionistic first-order logic can be transformed into another Kripke model with the full infinite binary tree as the Kripke frame such that both Kripke models force the same sentences. So with the previous result, there is a binary Kripke model K of HA such that K is not locally IΔ1.

NOTES ON THE DPRM PROPERTY FOR LISTABLE STRUCTURES

AbstractA celebrated result by Davis, Putnam, Robinson, and Matiyasevich shows that a set of integers is listable if and only if it is positive existentially definable in the language of arithmetic. We investigate analogues of this result over structures endowed with a listable presentation. When such an analogue holds, the structure is said to have the DPRM property. We prove several results addressing foundational aspects around this problem, such as uniqueness of the listable presentation, transference of the DPRM property under interpretation, and its relation with positive existential bi-interpretability. A first application of our results is the rigorous proof of (strong versions of) several folklore facts regarding transference of the DPRM property. Another application of the theory we develop is that it will allow us to link various Diophantine conjectures to the question of whether the DPRM property holds for global fields. This last topic includes a study of the number of existential quantifiers needed to define a Diophantine set.

"A Smack of Irrelevance" in Inconsistent Mathematics?


 
 
 Recently, some proponents and practitioners of inconsistent mathe- matics have argued that the subject requires a conditional with ir- relevant features, i.e. where antecedent and consequent in a valid conditional do not behave as expected in relevance logics —by shar- ing propositional variables, for example. Here we argue that more fine-grained notions of content and content-sharing are needed to ex- amine the language of (inconsistent) arithmetic and set theory, and that the conditionals needed in inconsistent mathematics are not as irrelevant as it is suggested in the current literature. 
 
 

Modal Structuralism with Theoretical Terms

In this paper, we aim to explore connections between a Carnapian semantics of theoretical terms and an eliminative structuralist approach in the philosophy of mathematics. Specifically, we will interpret the language of Peano arithmetic by applying the modal semantics of theoretical terms introduced in Andreas (Synthese 174(3):367–383, 2010). We will thereby show that the application to Peano arithmetic yields a formal semantics of universal structuralism, i.e., the view that ordinary mathematical statements in arithmetic express general claims about all admissible interpretations of the Peano axioms. Moreover, we compare this application with the modal structuralism by Hellman (Mathematics without numbers: towards a modal-structural interpretation. Oxford University Press: Oxford, 1989), arguing that it provides us with an easier epistemology of statements in arithmetic.

The impact of two different types of instructional tasks on students’ development of early algebraic thinking (El impacto de dos tipos diferentes de tareas instruccionales en el desarrollo del pensamiento algebraico temprano de los estudiantes)

ABSTRACT In this paper, we discuss the theoretical foundation and implementation of two alternative instructional courses that aimed to support the development of elementary school students’ early algebraic thinking. Both courses approached three basic algebra content strands: generalized arithmetic, functional thinking and modelling languages. The courses differed according to the characteristics of the tasks that were used. The first course involved ‘pure mathematical guided investigations’ emphasizing scaffolding steps in pure mathematics contexts. The second course focused on ‘applied open explorations’ underlining more open questions in applied everyday contexts. The courses were compared in respect to students’ learning outcomes. The findings, yielded from the analysis of pre-test and post-test data, indicated that the second course had better learning outcomes compared to the first.

An Overview on Learning Difficulty among School Children

Formal schooling in many other nations begins between ages 4 and 5, some research states that it starts earlier than this. The function of elementary schools throughout the world is to provide opportunities for children to acquire at least basic competencies in reading, writing, and computation. During middle childhood years, children are thought to be functioning developmentally at what Piaget termed the concrete and formal operational stages. Basic literacy as well as computational and conceptual skills is acquired at this phase. But some children may not be able to learn one or more of these skills as per their age. Their intellectual capacity and normal visual and physical abilities are unable to acquire one or more age appropriate language and /or arithmetic skills. They find it difficult to acquire the skill even when adequate learning opportunities are provided. These children have learning disabilities/ difficulties. Studies have revealed that nearly 10% of the childhood population has developmental language disorders of one type or the other and 8-10% of the school population has learning disability of one form or the other. Early intervention presupposes early identification. At present, there is no universally standardized screening procedure to guide referrals from schools. Interventions to rectify the issue should focus on developing and strengthening language and basic skills of reading, writing and arithmetic. In addition we should ensure that children should be allowed to “think” for them, to develop higher cognitive functioning is vital.

Generalized Realizability for Extensions of the Language of Arithmetic

Let L be an extension of the language of arithmetic, F be a class of number-theoretical functions. A notion of the V-realizability for L-formulas is defined in such a way that indexes of functions in V are used for interpreting the implication and the universal quantifier. It is proved that the semantics for L based on the V-realizability coincides with the classic semantics if and only if V contains all L-definable functions.

Absolute L-Realizability and Intuitionistic Logic

An absolute L-realizability of predicate formulas is introduced for all countable extensions L of the language of arithmetic. It is proved that the intuitionistic logic is not sound with this semantics.

The Quest of the Important in Mathematics Classroom

In this article, some ideas are exposed to take out profit of the properties of the straight line to know and to explore many interesting properties of functions as polynomials or rational functions. Some ideas are also presented to use the graphic calculators in a sequence that allows integrating the arithmetic language with the graphic and algebraic languages. This proposal has been applied in initial mathematic courses of future economists and engineers, obtaining interesting results related to the development of the graphic thought.

Quine’s Substitutional Definition of Logical Truth and the Philosophical Significance of the Löwenheim-Hilbert-Bernays Theorem

The Löwenheim-Hilbert-Bernays theorem states that, for an arithmetical first-order language L, if S is a satisfiable schema, then substitution of open sentences of L for the predicate letters of S results in true sentences of L. For two reasons, this theorem is relevant to issues relative to Quine’s substitutional definition of logical truth. First, it makes it possible for Quine to reply to widespread objections raised against his account (the lexicon-dependence problem and the cardinality-dependence problem). These objections purport to show that Quine’s account overgenerates: it would count as logically true sentences which intuitively or model-theoretically are not so. Second, since this theorem is a crucial premise in Quine’s proof of the equivalence between his substitutional account and the model-theoretic one, it enables him to show that, from a metamathematical point of view, there is no need to favour the model-theoretic account over one in terms of substitutions. The purpose of that essay is thus to explore the philosophical bearings of the Löwenheim-Hilbert-Bernays theorem on Quine’s definition of logical truth. This neglected aspect of Quine’s argumentation in favour of a substitutional definition is shown to be part of a struggle against the model-theoretic prejudice in logic. Such an exploration leads to reassess Quine’s peculiar position in the history of logic.